Multipole Coefficients Research Articles

Generalized fast multipole method

The fast multipole method (FMM) is a technique allowing the fast calculation of long-range interactions between N points in O(N) or O(N ln N) steps with some prescribed error tolerance. The FMM has found many applications in the field of integral equations and boundary element methods, in particular by accelerating the solution of dense linear systems arising from such formulations. Original FMMs required analytical expansions of the kernel, for example using spherical harmonics or Taylor expansions. In recent years, the range of applicability and the ease of use of FMMs has been extended by the introduction of black box [1] or kernel independent techniques [2]. In these approaches, the user only provides a subroutine to numerically calculate the interaction kernel. This allows changing the definition of the kernel with minimal change to the computer program. In this talk we will present a novel kernel independent FMM, which leads to diagonal multipole-to-local operators. This results in a significant reduction in the computational cost [1], in particular when high accuracy is needed. The approach is based on Cauchy's integral formula and the Laplace transform. We will present a numerical analysis of the convergence, methods to choose the parameters in the FMM given some tolerance, and the steps required to build a multilevel scheme from the single level formulation. Numerical results are given for benchmark calculations to demonstrate the accuracy as a function of the number of multipole coefficients, and the computational cost of the different steps in the method.

The close-limit approximation for black hole binaries with post-Newtonian initial conditions

The ringdown phase of a black hole formed from the merger of two orbiting black holes is described by means of the close-limit (CL) approximation starting from second-post-Newtonian (2PN) initial conditions. The 2PN metric of point-particle binaries is formally expanded in CL form and identified with that of a perturbed Schwarzschild black hole. The multipolar coefficients describing the even-parity (polar) and odd-parity (axial) components of the linear perturbation consistently satisfy the 2PN-accurate perturbative field equations. We use these coefficients to build initial conditions for the Regge–Wheeler and Zerilli wave equations, which we then evolve numerically. The ringdown waveform is obtained in two cases: head-on collision with zero-angular momentum, composed only of even modes, and circular orbits, for which both even and odd modes contribute. In a separate work, this formalism is applied to the study of the gravitational recoil produced during the ringdown phase of coalescing binary black holes.

Polarizable atomic multipole X-ray refinement: application to peptide crystals.

Recent advances in computational chemistry have produced force fields based on a polarizable atomic multipole description of biomolecular electrostatics. In this work, the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force field is applied to restrained refinement of molecular models against X-ray diffraction data from peptide crystals. A new formalism is also developed to compute anisotropic and aspherical structure factors using fast Fourier transformation (FFT) of Cartesian Gaussian multipoles. Relative to direct summation, the FFT approach can give a speedup of more than an order of magnitude for aspherical refinement of ultrahigh-resolution data sets. Use of a sublattice formalism makes the method highly parallelizable. Application of the Cartesian Gaussian multipole scattering model to a series of four peptide crystals using multipole coefficients from the AMOEBA force field demonstrates that AMOEBA systematically underestimates electron density at bond centers. For the trigonal and tetrahedral bonding geometries common in organic chemistry, an atomic multipole expansion through hexadecapole order is required to explain bond electron density. Alternatively, the addition of interatomic scattering (IAS) sites to the AMOEBA-based density captured bonding effects with fewer parameters. For a series of four peptide crystals, the AMOEBA-IAS model lowered R(free) by 20-40% relative to the original spherically symmetric scattering model.

Comparison of FDTD numerical computations and analytical multipole expansion method for plasmonics-active nanosphere dimers

This paper describes a comparative study of finite-difference time-domain (FDTD) and analytical evaluations of electromagnetic fields in the vicinity of dimers of metallic nanospheres of plasmonics-active metals. The results of these two computational methods, to determine electromagnetic field enhancement in the region often referred to as "hot spots" between the two nanospheres forming the dimer, were compared and a strong correlation observed for gold dimers. The analytical evaluation involved the use of the spherical-harmonic addition theorem to relate the multipole expansion coefficients between the two nanospheres. In these evaluations, the spacing between two nanospheres forming the dimer was varied to obtain the effect of nanoparticle spacing on the electromagnetic fields in the regions between the nanostructures. Gold and silver were the metals investigated in our work as they exhibit substantial plasmon resonance properties in the ultraviolet, visible, and near-infrared spectral regimes. The results indicate excellent correlation between the two computational methods, especially for gold nanosphere dimers with only a 5-10% difference between the two methods. The effect of varying the diameters of the nanospheres forming the dimer, on the electromagnetic field enhancement, was also studied.

Statistical dynamo theory: Mode excitation

We compute statistical properties of the lowest-order multipole coefficients of the magnetic field generated by a dynamo of arbitrary shape. To this end we expand the field in a complete biorthogonal set of base functions, viz. B= summation operator_{k}a;{k}(t)b;{k}(r) . The properties of these biorthogonal function sets are treated in detail. We consider a linear problem and the statistical properties of the fluid flow are supposed to be given. The turbulent convection may have an arbitrary distribution of spatial scales. The time evolution of the expansion coefficients a;{k} is governed by a stochastic differential equation from which we infer their averages a;{k} , autocorrelation functions a;{k}(t)a;{k *}(t+tau) , and an equation for the cross correlations a;{k}a;{l *} . The eigenfunctions of the dynamo equation (with eigenvalues lambda_{k} ) turn out to be a preferred set in terms of which our results assume their simplest form. The magnetic field of the dynamo is shown to consist of transiently excited eigenmodes whose frequency and coherence time is given by Ilambda_{k} and -1/Rlambda_{k} , respectively. The relative rms excitation level of the eigenmodes, and hence the distribution of magnetic energy over spatial scales, is determined by linear theory. An expression is derived for |a;{k}|;{2}/|a;{0}|;{2} in case the fundamental mode b;{0} has a dominant amplitude, and we outline how this expression may be evaluated. It is estimated that |a;{k}|;{2}/|a;{0}|;{2} approximately 1/N , where N is the number of convective cells in the dynamo. We show that the old problem of a short correlation time (or first-order smoothing approximation) has been partially eliminated. Finally we prove that for a simple statistically steady dynamo with finite resistivity all eigenvalues obey Rlambda_{k}<0 .

Broadband MLFMA With Plane Wave Expansions and Optimal Memory Demand

A broadband multilevel fast multipole algorithm (MLFMA) in 3D is presented based on plane wave expansions and diagonal translations on all division levels. The radiation and incoming wave patterns on all levels are presented by trigonometric polynomials, and optimal sampling rates and translator degrees are found and tabulated. On the super-wavelength levels the memory need and the computational cost are lowered more than by half compared to the traditional approach. On the sub-wavelength levels also a novel saving method for the field patterns is presented which is as efficient as saving the multipole series coefficients while all translations with the proposed method can still be carried out in the fast diagonal form. Also the direction dependence of the radiation and incoming wave patterns is lowered by half. The proposed broadband MLFMA has a very good error control on all division levels which we demonstrate by numerical testing.

Molecular shapes from small-angle X-ray scattering: extension of the theory to higher scattering angles

A low-resolution shape of a molecule in solution may be deduced from measured small-angle X-ray scattering I(q) data by exploiting a Hankel transform relation between the coefficients of a multipole expansion of the scattered amplitude and corresponding coefficients of the electron density. In the past, the radial part of the Hankel transform has been evaluated with the aid of a truncated series expansion of a spherical Bessel function. It is shown that series truncation may be avoided by analytically performing the radial integral over an entire Bessel function. The angular part of the integral involving a spherical harmonic kernel is performed by quadrature. Such a calculation also allows a convenient incorporation of a molecular hydration shell of constant density intermediate between that of the protein and the solvent. Within this framework, we determine the multipole coefficients of the shape function by optimization of the agreement with experimental data by simulated annealing.

Collective Multipole Expansions and the Perturbation Theory in the Quantum Three-Body Problem

The perturbation theory with respect to the potential energy of three particles is considered. The first-order correction to the continuum wave function of three free particles is derived. It is shown that the use of the collective multipole expansion of the free three-body Green function over the set of Wigner $D$-functions can reduce the dimensionality of perturbative matrix elements from twelve to six. The explicit expressions for the coefficients of the collective multipole expansion of the free Green function are derived. It is found that the $S$-wave multipole coefficient depends only upon three variables instead of six as higher multipoles do. The possible applications of the developed theory to the three-body molecular break-up processes are discussed.

Optical response of linear chains of metal nanospheres and nanospheroids

A semi-analytical method for computing the electric field surrounding a finite linear chain of metal nanospheres and nanospheroids is described. In treating chains or clusters of spheres, a common approach is to use the spherical-harmonic addition theorem to relate the multipole expansion coefficients between different spheres. A method is described here that avoids the use of spherical-harmonic addition theorems, which are not applicable to spheroidal chains. Simulations are given that illustrate the large field enhancements that can occur in the gaps between silver nanoparticles arising from plasmon resonances.

Estimates of multipolar coefficients for searching for cosmic ray anisotropies withnon-uniform or partial sky coverage

We study the possibility of extracting the multipolar moments of an underlyingdistribution from a set of cosmic rays observed with non-uniform or even partialsky coverage. We show that if the degree is assumed to be upper bounded byL, each multipolar moment can be recovered whatever the coverage,but with a variance increasing exponentially with the boundL if the coverage is zero somewhere. Despite this limitation, we show thepossibility of testing predictions of a model without any assumption onL by building an estimate of the covariance matrix seen through the exposure function.

Second and Third Order Electric Field Formulae Induced by Uniform ELF Magnetic Field in Eccentric Two Spherical Conductors

This paper describes analyses of electric fields in eccentric two spherical conductors induced by a uniform ELF magnetic field utilizing the equivalent multipole-moment method with re-expansion technique (RE-EMMM). Although an approximated formula has already been derived by Tarao in this situation, it has been limited to the first order approximation. Here, the authors mathematically derive a general expression of governing linear equation for unknown multipole and local coefficients. Furthermore, we derived the first, second, and third order electric field formulae by manually solving the linear equations. It is confirmed that the accuracy of the first order formula is almost the same to that of Tarao's. On the other hand, the accuracies of the second and third order formulae are confirmed to be more precise.

New Computational Procedure for Incorporating Computational Fluid Dynamics into Sonic Boom Prediction

A new computational approach for the inclusion of computational fluid dynamics flow solutions into predicting sonic boom signatures is developed. Using existing computational fluid dynamics tools, a near-field flow solution is obtained over the surface of a computational cylinder a certain distance away from the aircraft’s longitudinal axis. Near-field to far-field multipole-matching methodology is performed to calculate the corrected far field without calculating the multipole coefficients. The analytical derivation is provided and some results are presented and discussed.

A treecode for potentials of the form r −λ

It is known that the classical multilevel fast multipole method is a good technique for accelerating the iterative solution of integral equations with the Green's function kernel 1/r. However, few methods have been proposed for problems in molecular dynamics and computational chemistry where the Green's function is of the form r −λ, λ≥1. In this paper, we describe an efficient algorithm for computing potentials in which the integral equations have r −λ kernels. We propose a treecode algorithm which uses spherical harmonics to compute multipole coefficients that are used to evaluate these potentials. The key idea in this algorithm is the use of ultraspherical polynomials to represent r −λ in a manner analogous to the use of Legendre polynomials for the expansion of the Coulomb potential r −1. We exploit the relationship between ultraspherical and Legendre polynomials to develop a natural generalization of the multipole expansion theorem which was used in the multilevel fast multipole method. This theorem is used along with a hierarchical scheme to compute potentials of the form r −λ. The complexity of the algorithm is O(p 3 Nlog N) where N is the number of interacting particles and p is a small constant. We discuss the advantages of our algorithm over existing single level methods based on Cartesian coordinate expansion schemes. The efficiency of the algorithm is illustrated with numerical experiments.

Tensor microwave background fluctuations for large multipole order

We present approximate formulas for the tensor BB, EE, TT, and TE multipole coefficients for large multipole order $\ensuremath{\ell}$. The error in using the approximate formula for the BB multipole coefficients is less than cosmic variance for $\ensuremath{\ell}g10$. These approximate formulas make various qualitative properties of the calculated multipole coefficients transparent: specifically, they show that, whatever values are chosen for cosmological parameters, the tensor EE multipole coefficients will always be larger than the BB coefficients for all $\ensuremath{\ell}g15$, and that these coefficients will approach each other for $\ensuremath{\ell}\ensuremath{\ll}100$. These approximations also make clear how these multipole coefficients depend on cosmological parameters.

2D Magnetic Design and Optimization of a 88-mm Aperture 15 T Dipole for NED

The Next European Dipole (NED) activity supported by the European Union aims at the development of a high-performance Nb3Sn conductor (Jc = 1500 A/mm2 @ 15 T, 4.2 K) in collaboration with European industry and at the design of a high- field dipole magnet making use of this conductor. In the framework of the NED collaboration which coordinates the activity of several institutes, CERN has contributed to the electromagnetic design study of a cos - thetas, layer-type superconducting dipole with an 88 mm aperture that is able to reach 15 T at 4.2 K. Part of the optimization process was dedicated to the reduction of the multipole coefficients so as to improve field quality while keeping an efficient peak-field to main-field ratio. In this paper, we present the optimization of the coil cross-section and of the shape of the iron yoke to reduce saturation-induced field errors during ramp. The effects of persistent magnetization currents are also estimated and different methods to compensate persistent-current-induced field distortions are presented.

Dose mapping using multipole moments

Conventional dose distribution study for gamma irradiation facility involves the use of isodose maps. The establishment of these maps requires the measurement of the absorbed dose in many points, which makes the task expensive experimentally and very long by simulation. However, a lack of points of measurement can distort the dose distribution cartography. To overcome these problems, we present in this paper a mathematical method to describe the dose distribution in air. This method is based on the multipole expansion in spherical harmonics of the photon flux emitted by the gamma source. The determination of the multipole coefficients of this development allows the modeling of the radiation field around the gamma source.

Multipolar representation of protein structure.

BackgroundThat the structure determines the function of proteins is a central paradigm in biology. However, protein functions are more directly related to cooperative effects at the residue and multi-residue scales. As such, current representations based on atomic coordinates can be considered inadequate. Bridging the gap between atomic-level structure and overall protein-level functionality requires parameterizations of the protein structure (and other physicochemical properties) in a quasi-continuous range, from a simple collection of unrelated amino acids coordinates to the highly synergistic organization of the whole protein entity, from a microscopic view in which each atom is completely resolved to a "macroscopic" description such as the one encoded in the three-dimensional protein shape.ResultsHere we propose such a parameterization and study its relationship to the standard Euclidian description based on amino acid representative coordinates. The representation uses multipoles associated with residue Cα coordinates as shape descriptors. We demonstrate that the multipoles can be used for the quantitative description of the protein shape and for the comparison of protein structures at various levels of detail. Specifically, we construct a (dis)similarity measure in multipolar configuration space, and show how such a function can be used for the comparison of a pair of proteins. We then test the parameterization on a benchmark set of the protein kinase-like superfamily. We prove that, when the biologically relevant portions of the proteins are retained, it can robustly discriminate between the various families in the set in a way not possible through sequence or conventional structural representations alone. We then compare our representation with the Cartesian coordinate description and show that, as expected, the correlation with that representation increases as the level of detail, measured by the highest rank of multipoles used in the representation, approaches the dimensionality of the fold space.ConclusionThe results described here demonstrate how a granular description of the protein structure can be achieved using multipolar coefficients. The description has the additional advantage of being immediately generalizable for any residue-specific property therefore providing a unitary framework for the study and comparison of the spatial profile of various protein properties.

A FAST BOUNDARY ELEMENT METHOD FOR UNDERWATER ACOUSTICS

A fast algorithm for the boundary element method is developed to handle problems in underwater acoustics. The algorithm employs the multipole and local expansions to approximate far-field potentials, and exploits the discrete convolution nature of mapping multipole to local expansions to accelerate the potential evaluation process. The speedup in the solution process is achieved by fast Fourier transform on the multipole and local expansion coefficients on a regular grid. The method is demonstrated by a three-dimensional underwater acoustics scattering problem, and it is found to achieve accurate results with relatively low order of expansion.

Multipole Vector Anomalies in the First‐YearWMAPData: A Cut‐Sky Analysis

We apply the recently defined multipole vector framework to the frequency-specific first-year WMAP sky maps, estimating the low-l multipole coefficients from the high-latitude sky by means of a power equalization filter. While most previous analyses of this type have considered only heavily processed (and foreground-contaminated) full-sky maps, the present approach allows for greater control of residual foregrounds, and therefore potentially also for cosmologically important conclusions. The low-l spherical harmonics coefficients and corresponding multipole vectors are tabulated for easy reference. Using this formalism, we re-assess a set of earlier claims of both cosmological and non-cosmological low-l correlations based on multipole vectors. First, we show that the apparent l=3 and 8 correlation claimed by Copi et al. (2004) is present only in the heavily processed map produced by Tegmark et al. (2003), and must therefore be considered an artifact of that map. Second, the well-known quadrupole-octopole correlation is confirmed at the 99% significance level, and shown to be robust with respect to frequency and sky cut. Previous claims are thus supported by our analysis. Finally, the low-l alignment with respect to the ecliptic claimed by Schwarz et al. (2004) is nominally confirmed in this analysis, but also shown to be very dependent on severe a-posteriori choices. Indeed, we show that given the peculiar quadrupole-octopole arrangement, finding such a strong alignment with the ecliptic is not unusual.

Magnetic anisotropy in the cubic LavesREFe2 intermetallic compounds

In the past, the Callen–Callen (1965 Phys. Rev. 139 A455–71; 1966 J. Phys. Chem. Solids27 1271–85) model has been highly successful in explaining the origin andtemperature dependence of the magneto-crystalline anisotropy in manymagnetic compounds. Yet, despite their high ordering temperatures of∼650 K, the Callen–Callen model has proved insufficient for theREFe2 compounds. In this paper, we show that it is possible to replicate the values of the phenomenologicalparameters K1, K2, andK3 givenby Atzmony and Dariel (1976 Phys. Rev. B 13 4006–14), by extending the Callen–Callen model to secondorder in HCF. In particular, explanations are provided for (i) the unexpected changes in sign ofK1 and K2 in HoFe2 and DyFe2, respectively, and (ii) the origin and behaviour of theK3 term. In addition, it is demonstrated that higher order terms are required, and thatK4 exceedsK3 at low temperatures.Revised estimates of K1, K2,K3,K4,and K5 are given. Finally, an alternative ‘multipolar’ approach to the problem of magneticanisotropy is also provided. It is shown that the latter confers significant advantages overthe older phenomenological method. In particular, all the multipolar coefficients (, N = 4, 6, 8, 10, 12) decrease monotonically with increasing temperature, with decreasing faster than etc. These observations are in accord with expectations based on the originalCallen–Callen model.